There is only one real formal way to say hello in Mandarin Chinese, and whilst you can translate literal meanings from many English greetings the only one that is used is 你好 (nǐ hǎo). Below are some introductory sentences, greetings, how to introduce yourself and how to say bye. There is a lot to take in here, but I hope it useful. Please suggest ways to continue conversations below in the comments.

I am...
我是...
wǒ​ shì​...

How are you?
你好吗
nǐ hǎo ma?

How are you doing?
你怎么样?
Nǐ zěnme yàng?

How are you today?
你今天好吗？​
nǐ jīntiān hǎo ma?

Did u sleep well last night?
昨晚睡得睡的好吗?
zúo​ wǎn shùi de hǎo ma?

I am Good
我很好
wǒ hěn hǎo

And You?
你呢？
nǐ ne?

Good Thank You
好谢谢你.
hǎo xiè​xie​ nǐ.

Nice to meet you!
幸会
Xìng huì!

The weather is nice today, isn't it?
今天的天气真好,是不是?
jīn​tiān​ de tiān​qì​ zhēn​ hǎo​, shì​bù​shì​?

Saying Hello at Different Times of Day


Good Morning
早上好
zǎo​shang​hǎo​

or

早安
zǎo ān
(literally 'early peace')
- mainly used in Taiwan

Good Afternoon
下午好
xià​wǔ​hǎo​

or

午安
Wǔ ān
- mainly used in Taiwan

Good Night
晚上好
Wǎnshàng hǎo

or

晚安
Wǎn ān
- mainly used in Taiwan


Saying Goodbye in Mandarin


I have to go now.
我必须走了。
wǒ bì xū zǒu le.

Goodbye
再见
zài​jiàn​

Byebye
拜拜
Bàibài

See You Later
回头见 jiàn
huí​tóu​

or

待会儿见
dài huìr jiàn

I need to go to sleep
我需要睡觉
wǒ​ xū​yào​ shuì​jiào​

See you Tomorrow
明天见
míng​tiān​ jiàn